In this section we illustrate our theoretical results by some financial applications.
Namely, we consider the problem of pricing Bermudan options.
The pricing of American-style options is one of the most challenging problems in
computational finance, particularly when more than one factor affects the option values.
Simulation based methods have become increasingly attractive
compared to
other numerical methods as the dimension of the problem increases. The reason for this is
that the convergence rates of simulation based methods are generally independent of the number
of state variables.
In the context of our paper we consider the so called parametric approximation algorithms
.
In essence, these algorithms represent the optimal stopping sets \( \mathcal {S}^{*}_{k} \)  by a finite numbers of parameters and
then find the Bermudan option price by maximizing, over the parameter space, a Monte Carlo approximation of the corresponding value function.
The important question here is wether on can parametrize the optimal stopping region \( \mathcal {S}^{*} \)  by a finite dimensional set of parameters, i.e. \( \mathcal {S}^{*}=\mathcal {S}(\theta ), \, \theta \in \Theta , \)  where \( \Theta \)  is a compact finite dimensional set. It turns out that that
this is possible in many situations (see [1]}). The assumption (REF ) and (REF ) are then automatically fulfilled with arbitrary small \( \rho >0. \)
